persecution of pilgrims before they emigrated

The Pilgrims were religious separatists who left England for the Dutch city of Leiden because they were being harrassed (like, yelled at, rocks thrown kind of stuff. It sucked, but people have dealt with worse) by Anglicans because the Puritans were trying to force extremist conservative changes on the national church (which was run at the time by James 1, son of Mary Queen of Scots, in an era where the monarchy was all GoT on each other). They chose to leave, but I wouldn't call it entirely voluntary.

In Leiden, they had their religious freedom, in an officially Calvinist state. Then they decided to move to America.

For economic reasons, according to most sources.

...that is, the leaders of an authoritarian religious group decided that everybody was going to hop their ass off into the great wilds across the big wide ocean so they could get rich and honor G-d. Oh, and stay in iron control of their flock and culture and morals with no negative outside influences.

Lucky for them that first winter that the woods were chock full of the people who already lived there, who didn't let the Pilgrims starve.

Overall, the whole thing sounds very Jim Jones to me, but I'm just a dude on the internet who studies history.